How to find the determinant of this matrix?

In summary, the speakers discuss ways to simplify the calculation of the determinant for a specific matrix. One suggests reducing the matrix to a 2x2 matrix using the Chio method, while another suggests using row operations or the polynomial method. They also mention that exchanging rows changes the sign of the determinant and suggest using induction to prove the determinant formula.
  • #1
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I think you all can see that ##a_{(i+1,j+1)} = a_{i,j} + a_{i+1,j} + a_{i,j+1}##

Now the determinant always give me problem. I have and idea to reduce this matrix by Chio to a 2x2 matrix and find the determinant of this 2x2.

Put i was not able to see any pattern to find what how the 2x2 matrix would be (beside symmetric)

Any tips?
 
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  • #2
I would first try to write the matrix as a product of two simpler matrices, because the construction rule is similar to matrix multiplication. If this would be too complicated, I'd try the polynomial method: ##\det = \sum_{\sigma\in S_n}(-1)^n \ldots##
 
  • #3
I want to suggest using row operations to reduce the matrix to something more manageable.

What could be helpful is the following.

Adding or subtracting any two rows of a matrix does not change the determinant.
Exchanging two rows of a matrix changes the sign of the determinant.
 
  • #4
Another idea is to prove by induction that the determinant equals ##2^{(n^2-n)/2}=2^{\binom n 2}##.
 
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Related to How to find the determinant of this matrix?

1. What is the definition of a determinant?

A determinant is a numerical value that is calculated from a square matrix. It is used to determine certain properties of the matrix, such as whether it is invertible or singular.

2. How do I find the determinant of a 2x2 matrix?

To find the determinant of a 2x2 matrix, you can use the formula ad - bc, where a, b, c, and d are the elements of the matrix. Simply multiply the elements in the top left and bottom right positions, and then subtract the product of the top right and bottom left elements.

3. What is the process for finding the determinant of a larger matrix?

For a larger matrix, you can use a method called expansion by minors. This involves breaking the matrix down into smaller submatrices and calculating the determinants of those submatrices. You can then use these determinants to find the determinant of the original matrix.

4. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the number of row swaps that are required to reduce the matrix to its upper triangular form. An even number of row swaps will result in a positive determinant, while an odd number will result in a negative determinant.

5. How is the determinant used in real-world applications?

The determinant is used in a variety of fields, including physics, engineering, and economics. It is used to solve systems of linear equations, calculate areas and volumes, and determine the stability of a system. In economics, it is used to analyze input-output models and to calculate the elasticity of demand.

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